{\setlength{\arraycolsep}{1.5pt}
\begin{array}{rcl}
p_0 & := & 1 \\
q_0 & := & \displaystyle\frac{p_0}{\left\|p_0\right\|} = \frac{1}{\sqrt{2}} \\
p_1 & := & \displaystyle x - \left<x,q_0\right>q_0 = x - \frac{1}{\sqrt{2}}\int_{-1}^1\frac{1}{\sqrt{2}}xdx = x \\ 
q_1 & := & \displaystyle\frac{p_1}{\left\|p_1\right\|} = \sqrt{\frac{3}{2}}x \\
p_2 & := & x^2 - \left<x^2,q_1\right>q_1 - \left<x^2,q_0\right>q_0 = \\
 & = & \displaystyle x^2-\frac{3}{2}x\int_{-1}^{1}\frac{3}{2}x^3dx-\frac{1}{\sqrt{2}}\int_{-1}^{1}\frac{1}{\sqrt{2}}x^2dx = \\
 & = & \displaystyle x^2-\frac{1}{3} \\
q_2 & := & \displaystyle\frac{p_2}{\left\|p_2\right\|} = \sqrt{\frac{45}{8}}\left(x^2-\frac{1}{3}\right) \\
p_3 & := & x^3 - \left<x^3,q_2\right>q_2 - \left<x^3,q_1\right>q_1 - \left<x^3,q_0\right>q_0 = \\
 & = & \displaystyle\ldots = x^3 - \frac{3}{5}x \\
q_3 & := & \displaystyle\frac{p_3}{\left\|p_3\right\|} = \sqrt{\frac{175}{8}}\left(x^3-\frac{3}{5}x\right)
\end{array}}